I can tell you more of what isn't the answer than what is, unfortunately.

It's certainly nothing to do with the Catholic Church. No Roman calendar ever began on the winter solstice. January 1st has been New Year's Day since at least 153 BC, and before then the Romans started their year in March, at the beginning of both the agricultural year and the military campaigning season. The early history of the Roman calendar is quite murky, but in origin it was lunar, beginning months with the new moon. Exactly how the Roman calendar developed over the Republic up to Caesar's reforms in 45 BC is not really relevant here, but the point is that the Roman never marked time using the solstices, although they did eventually recognise that a lunar calendar doesn't match the solar year. Why they never shunted January 1st back a week to coincide with the solstice I don't know, but presumably you can't do that without doing serious violence to the timing of all the other festivals in the religious calendar. That was probably a higher concern to the Romans who, as I'm about to explain, where not actually that bothered about the religious aspects of the sun's movements anyway.

To (sort of) address the second part of your question, the Romans didn't worship the sun at the solstices. In surviving calendars the festivals of Sol are in 8/9th August (also 28th, together with Luna, the moon) 19th and 22nd October and 11th December, plus the late (and questionable) addition of 25th December. None of these are directly connected to the movements of the sun, although I suppose the December festivals are close.